Brandenburg University of Technology at Cottbus

We report on the results of an investigation into the application of three Petri net approaches for the modelling and analysis of biochemical networks: discrete, continuous and stochastic Petri nets. We mimic the modelling and analysis steps, which have been proposed in [1] for an extended ERK pathway model. We can confirm that analyses based on a discrete Petri net model of the system can be used to derive the sets of initial concentrations required by the corresponding continuous ordinary differential equation model which is equivalent to a continuous Petri net, and no other initial concentrations produce meaningful steady states. Furthermore, we show how a stochastic Petri net model with discrete levels for the concentration of a species can be applied to perform probabilistic model checking, as proposed in [2]. 1 Motivation This paper is based on the Technical report “From Petri Nets to Differential Equations an Integrative Approach for Biochemical Network Analysis” by Gilbert and Heiner [1]. Here, we want to employ the proposed discrete and continuous Petri net modelling and analysis techniques to another extended ERK pathway model, which includes more species and reactions. In addition, we will employ a third Petri net formalism, which can be useful for modelling signalling transduction pathways: stochastic Petri nets. Inspired by the work of Calder et. al [2], we will combine the concept of discrete levels for the concentration of a species with the well-known Petri net technique of different colours for different types of tokens for this purpose. To avoid the repetition of all detailed facts about the discrete and the continuous Petri net approach, we assume the knowledge of [1] and the basic mechanisms of the introduced modelling and analysis techniques. In general, a discrete Petri net always describes the set of all possible behaviours of the system. If the discrete Petri net is bounded, we can identify a 2 Lehrack, Sebastian particular simulation run with a run through the computable reachability graph. In this approach the duration of a transition is neglected; here we are only interested in causal relations. Thus, the discrete Petri net model itself implicitly contains all possible time behaviours. When we define a particular duration for each transition/reaction and introduce a continuous token flow, we obtain a determined dynamic behaviour for our continuous Petri net. The concentration of a particular species will have the same value in each simulation run at a particular time point. Besides this deterministic continuous approach, there is a further commonly used method to describe the time behaviour of dynamic biological system: the non-deterministic stochastic approach. Gillespie [3] pointed out that a stochastic individual molecular based approach as a kind of random-walk process better describes the real physical world rather than a deterministic approach such as ODEs which are an average behaviour of the population of molecules. The previous paper already recommended to model the qualitative, discrete model before the quantitative continuous one. In this work we integrate stochastic model in the whole modelling process. One of the major profits of the added stochastic approach is the opportunity to carry out probabilistic model checking, so we will be able to extend temporal logic queries with probabilities. Figure 1 shows the relations between our three approaches and the corresponding analysis techniques. Fig. 1. Possible modelling process of discrete, continuous and stochastic Petri nets This paper is organized as follows. The next section provides an overview on the biochemical context on hand and introduces the running example. Afterwards, we demonstrate the application of the modelling and analysis techniques in section 3 and 4, as proposed in [1]. Section 5 is devoted to the stochastic viewpoint. We conclude with a summary and outlook on intended further research directions. Three Petri net approaches for Biochemical Network Analysis 3 2 Biochemical Context To mimic and extend the proposed analysis techniques of [1] we choose an extended ERK pathway model, which is published in [4]. In the following we will use the notion extended ERK pathway model to differ this model from the ERK pathway model, which is introduced in [5] and analysed in the previous paper [1]. Furthermore, we will use the names of the species as they are used in [4]. In general, the ERK pathway (also called Ras/Raf, or Raf-1/MEK/ERK pathway) is a ubiquitous pathway that conveys cell division and differentiation signals from the cell membrane to the nucleus. Ras is activated by an external stimulus, via one of many growth factor receptors; it then binds to and activates RasGTP to become Rafx, or activated Raf, which in turn activates MAPK/ERK Kinase (MEK) which in turn activates Extracellular signal Regulated Kinase (ERK). This cascade (RasGTP → Raf → MEK → ERK) of protein interaction controls cell differentiation, the effect being dependent upon the activity of ERK. In our extended ERK pathway model we consider a certain concentration level of RasGTP as input signal and a certain concentration level of ERKPP (activated ERK) as output signal. 3 The Discrete Approach In this section we apply place/transition Petri nets to model the pathway of interest, and interpret them in the standard way. The software tools which have been used in this section are: for modelling – Snoopy [6], and for analysis – the Integrated Net Analyser (INA) [7], and the Model Checking Kit [8]. 3.1 Qualitative Modelling We create a place/transition Petri nets, see Figure 2. The Petri net model has been derived manually from the set of the ODEs, given in [4]. Places represented as circles stand for the states of a protein or protein complex and are labelled with the corresponding name; complexes are indicated by an underscore “ ” between the protein names. For example, MEK and Rafx are proteins, and MEK Rafx is a protein complex formed from MEK and Rafx. A suffix -P or -PP denotes a single or double phosphorylated protein, for example MEKP and ERKPP. The terms Phase1, Phase2 and Phase3 stand for particular phosphatases, which are required for the dephosphorylation of Rafx, MEKP, MEKPP, ERKP and ERKPP. In the extended ERK pathway model under consideration there are 22 proteins or complexes; a discrete concentration is associated with each protein or complex. In the case of the qualitative model, these concentrations can be thought of as being ‘high’ or ‘low’ (present or absent). Transitions represented as rectangles stand for reactions, with reversible reactions being indicated by a rectangle containing an inner rectangle (hierarchical transitions). In this extended ERK pathway model, reactions comprise protein 4 Lehrack, Sebastian complexation and decomplexation events, often accompanied by phosphorylation or dephosphorylation. Reactions can be one-way, indicated by single-headed arrows, or reversible, indicated by double-headed arrows on the top level of the net. For example, MEK and Rafx combine in a forwards reaction to form MEK Rafx which can disassociate in a backwards reaction into MEK and Rafx. In this qualitative model, k1, k2, . . . stand for reaction labels. Raf RasGTP Raf_RasGTP